The surge in cars predicted by the auto maker would likely tax its existing network of about 150 Tesla service centers

As Tesla Inc. begins a pivotal launch of its first mass-market car, the company said it plans to triple its capacity to repair vehicles, adding 1,400 technicians, dozens of new service centers and hundreds of maintenance vans that can be dispatched to an owner’s home.

The Silicon Valley auto maker started production last week of the Model 3, a $35,000 sedan that Chief Executive Elon Musk is betting will boost Tesla’s production to 500,000 vehicles next year from 84,000 last year. The surge of cars—coupled with any...